Problem: The bacteria in a Petri dish culture are self-duplicating at a rapid pace. The relationship between the elapsed time $t$, in minutes, and the number of bacteria, $B(t)$, in the Petri dish is modeled by the following function. B ( t ) = 100 ⋅ 2 t 10 B(t)=100 \cdot 2\^{\frac{t}{10}} How many minutes will it take for the culture to achieve $10{,}000$ bacteria? Round your answer, if necessary, to the nearest hundredth.
Explanation: Thinking about the problem We want to know how many minutes, $t$, it will take for the number of bacteria in the culture, $B(t)$, to reach $10{,}000$. So, we need to find the value of $t$ for which $B(t)=10{,}000$. Substituting $10{,}000$ in for $B(t)$ in the function gives us the following equation. 10,000 = 100 ⋅ 2 t 10 10{,}000=100 \cdot 2\^{\frac{t}{10}} Solving the equation We can solve the equation as shown below. 100 ⋅ 2 t 10 2 t 10 t 10 t = 10000 = 100 = log 2 ( 100 ) = 10 log 2 ( 100 ) \begin{aligned}100\cdot2\^{\frac{t}{10}}&=10000\\\\ 2\^{\frac{t}{10}}&=100\\\\ \dfrac{t}{10}&=\log_2(100)\\\\ t&=10\log_2(100)\\\\ \end{aligned} Changing the base to approximate the solution Since most calculators only calculate logarithms in base $10$ and base $e$, let's change the base. [What is the change of base rule?] $\begin{aligned}t&=10\log_2(100)\\\\ &=10\cdot\left(\dfrac{\log(100)}{\log(2)}\right)\\\\ &\approx 66.44 \end{aligned}$ The culture will reach $10{,}000$ bacteria after $66.44$ minutes.